Unlikely you'll get a comprehensive answer to such a general question here. Have you tried wikipedia/google?

That is a good suggestion! Also there is no one unique set of mathematical equations or physical principles defining "String theory". It is easier to describe what String is as a research program.

To some extent the String Program is defined by the annual Strings conference, which was excellent this year! Very well organized, with high quality video of all 40 talks currently available online.

David Gross is usually the person chosen to give a kind of "State of the Program" talk at the annual Strings conference, and this year (as he often does) he gave the final talk of the conference, titled "Vision and Outlook". You might want to watch the video!

As I recall he says things like: There are some missing pieces. String is more of a "framework" than a definite theory. "We" (the string community) do not know what the theory actually is. He presents 3 main questions he hopes will be answered in his lifetime. It is an upbeat, optimistic and inspiring talk---and rightly titled "Vision and Outlook."

This talk is similar to ones that Gross gave at other Strings conferences (madrid 2007, geneva 2008, rome 2009, texas 2010, sweden 2011) but I found it, if anything, more hopeful and inspiring this time. One should definitely watch it to get an idea of what String framework/community/program is about--what the researchers are interested in, what they are doing, what the outlook is, etc.

And there is, so far, no one unique theory. There is, Gross stressed in his talk, a framework of ideas within which one can construct many theories (so far none of them completely agreeing with the real world.)

I read somewhere that the mathematical proving behind it is very logical that's why I'm asking for a mathematical explanation.

Sure. After all, string theory is a mathematical framework. But that is different from the OP's question of whether string theory can be somehow shown to be correct via mathematical proof. The two main problems with this: 1) it's a physical theory and so it must be subject to experimental verification and 2) as Marcus has mentioned, the core principles of the theory are unknown.

The core principles aren't known - it's important to realise that people dont yet know how to define `string theory'.

I think sashickman was just saying that with perturbative string theory equations CAN actually be written down. That's not to say that's the definition of string theory or that it passes physical checks - for e.g. does the perturbative series add up in a sensible way - that is is perturbative super symmetric string theory Borel summable? (the Bosonic string theory isn't).